Hypo-elasticity, Cauchy-elasticity, corotational stability and monotonicity in the logarithmic strain

Abstract

We combine the rate-formulation for the objective, corotational Zaremba-Jaumann rate \begin{align} \frac{{\rm D}^{\rm ZJ}}{{\rm D} t} [σ] = \mathbb{H}^{\rm ZJ}(σ).D, \qquad D = {\rm sym} {\rm D} v\,, \end{align} operating on the Cauchy stress $\ddot{I}ƒ$, the Eulerian strain rate $D$ and the spatial velocity $v$ with the novel \enquote{corotational stability postulate} (CSP)\begin{equation} \Bigl\langle \frac{{\rm D}^{\rm ZJ}}{{\rm D} t}[σ], D \Bigr\rangle > 0 \qquad \forall \, D\in{\rm Sym}(3)\setminus\{0\} \end{equation} to show that for a given isotropic Cauchy-elastic constitutive law $B\mapsto \ddot{I}ƒ(B)$ in terms of the left Cauchy-Green tensor $B=F{F}^T$, the induced fourth-order tangent stiffness tensor ${\mathbb{H}}^{\mathrm{ZJ}}(\ddot{I}ƒ)$ is positive definite if and only if for $\hat{\ddot{I}}ƒ(\log B):=\ddot{I}ƒ(B)$, the strong monotonicity condition (TSTS-M${}^{++}$) in the logarithmic strain is satisfied. Thus (CSP) implies (TSTS-M^{++}) and vice-versa, and both imply the invertibility of the hypo-elastic material law between the stress and strain rates given by the tensor ${\mathbb{H}}^{\mathrm{ZJ}}(\ddot{I}ƒ)$. The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M^{++}) a far reaching generality. It is conjectured that this characterization of (CSP) holds for a large class of reasonable corotational rates. The result for the logarithmic rate is based on a novel chain rule for corotational derivatives of isotropic tensor functions.